Mereologies in Computing Science Uppsala: Thursday, 11 November 2010 - - PowerPoint PPT Presentation

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Mereologies in Computing Science

Uppsala: Thursday, 11 November 2010

Dines Bjørner

October 30, 2010, 15:06, Uppsala Seminar, 11 Nov. 2010 c Dines Bjørner 2010, Fredsvej 11, DK–2840 Holte, Denmark

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• 1. Abstract
• 1. Abstract

In this talk we solve the following problems:

• we give a formal model of a large class of mereologies,

– with simple entities modelled as parts – and their relations by connectors;

• we show that that class applies to a wide variety of societal

infrastructure component domains;

• we show that there is a class of CSP channel and process structures

that correspond to the class of mereologies where – mereology parts become CSP processes and – connectors become channels; – and where simple entity attributes become process states.

c Dines Bjørner 2010, Fredsvej 11, DK–2840 Holte, Denmark Uppsala Seminar, 11 Nov. 2010 October 30, 2010, 15:06

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• 1. Abstract
• We have yet to prove to what extent the models satisfy

– the axiom systems for mereologies of, for example, (Casati&Varzi 1999) – and a calculus of individuals (Bowman&Clarke 1981).

• Mereology is the study, knowledge and practice of part-hood

relations: – of the relations of part to whole and – the relations of part to part within a whole.

• By parts we shall here understand simple entities — of the kind

illustrated in this talk.

October 30, 2010, 15:06, Uppsala Seminar, 11 Nov. 2010 c Dines Bjørner 2010, Fredsvej 11, DK–2840 Holte, Denmark

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• 1. Abstract
• Manifest simple entities of domains

– are either continuous (fluid, gaseous) – or discrete (solid, fixed), and if the latter, then ∗ either atomic ∗ or composite. – It is how the sub-entities of a composite entity ∗ are “put together” ∗ that “makes up” a mereology of that composite entity — at least such as we shall study the mereology concept.

• In this talk we shall study some ways of modelling the mereology of

composite entities.

c Dines Bjørner 2010, Fredsvej 11, DK–2840 Holte, Denmark Uppsala Seminar, 11 Nov. 2010 October 30, 2010, 15:06

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• 1. Abstract
• One way of modelling mereologies is using

– sorts, – observer functions and – axioms (McCarthy style),

• another is using CSP.

October 30, 2010, 15:06, Uppsala Seminar, 11 Nov. 2010 c Dines Bjørner 2010, Fredsvej 11, DK–2840 Holte, Denmark

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• 2. Introduction
• 2. Introduction

2.1. Physics and Societal Infrastructures

2.1.1. Physics

• Physicists study that of nature which can be measured

– within us, – around us and – between ‘within’ and ‘around’!

• To make mathematical models of physics phenomena,

– physics has helped develop and uses mathematics, – notably calculus and statistics.

c Dines Bjørner 2010, Fredsvej 11, DK–2840 Holte, Denmark Uppsala Seminar, 11 Nov. 2010 October 30, 2010, 15:06

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• 2. Introduction 2.1. Physics and Societal Infrastructures 2.1.1. Physics
• Domain engineers primarily studies societal infrastructure

components which can be – reasoned about, – built and – manipulated by humans.

• To make domain models of infrastructure components, domain

engineering makes use of – formal specification languages, – their reasoning systems: formal testing, model checking and verification, and – their tools.

October 30, 2010, 15:06, Uppsala Seminar, 11 Nov. 2010 c Dines Bjørner 2010, Fredsvej 11, DK–2840 Holte, Denmark

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• 2. Introduction 2.1. Physics and Societal Infrastructures 2.1.2. In Nature

2.1.2. In Nature

• Physicists turns to algebra in order to handle structures in

nature. – Algebra appears to be useful in a number of applications, to wit: ∗ the abstract modelling of chemical compounds. – But there seems to be many structures in nature ∗ that cannot be captured in a satisfactory way by mathematics, including algebra ∗ and when captured in discrete mathematical disciplines such as sets, graph theory and combinatorics · the “integration” of these mathematically represented — structures · with calculus (etc.) — becomes awkward; · well, I know of no successful attempts.

c Dines Bjørner 2010, Fredsvej 11, DK–2840 Holte, Denmark Uppsala Seminar, 11 Nov. 2010 October 30, 2010, 15:06

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• 2. Introduction 2.1. Physics and Societal Infrastructures 2.1.2. In Nature
• Domain engineers turns to discrete mathematics —

– as embodied in formal specification languages – and as “implementable” in programming languages — in order to handle structures in societal infrastructure components.

• These languages allow

– (a) the expression of arbitrarily complicated structures, – (b) the evaluation of properties over such structures, – (c) the “building & demolition” of such structures, and – (d) the reasoning over such structures.

• They also allow the expression of dynamically varying structures —

– something mathematics is “not so good at” !

October 30, 2010, 15:06, Uppsala Seminar, 11 Nov. 2010 c Dines Bjørner 2010, Fredsvej 11, DK–2840 Holte, Denmark

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• 2. Introduction 2.1. Physics and Societal Infrastructures 2.1.2. In Nature
• But the specification languages have two problems:

– (i) they do not easily, if at all, ∗ handle continuity, that is, they do not embody calculus, ∗ or, for example, statistical concepts, etc., and – (ii) they handle ∗ actual structures of societal infrastructure components ∗ and attributes of atomic and composite entities of these – – usually by identical techniques – thereby blurring what we think is an important distinction.

c Dines Bjørner 2010, Fredsvej 11, DK–2840 Holte, Denmark Uppsala Seminar, 11 Nov. 2010 October 30, 2010, 15:06

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• 2. Introduction 2.2. Structure of This Talk

2.2. Structure of This Talk

• The rest of the talk is organised as follows.
• First we give a first main, a meta-example,

– of syntactic aspects of a class of mereologies.

• We informally show that the assembly/unit structures indeed

model structures of a variety of infrastructure components.

• Then we discuss concepts of atomic and composite simple entities.

October 30, 2010, 15:06, Uppsala Seminar, 11 Nov. 2010 c Dines Bjørner 2010, Fredsvej 11, DK–2840 Holte, Denmark

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• 2. Introduction 2.2. Structure of This Talk
• We then “perform”

– the ontological trick of mapping the assembly and unit entities – and their connections – exemplified in the first main meta-example – into CSP processes and channels, respectively — – the second and last main — meta-example and now ∗ of semantic aspects of a class of mereologies.

c Dines Bjørner 2010, Fredsvej 11, DK–2840 Holte, Denmark Uppsala Seminar, 11 Nov. 2010 October 30, 2010, 15:06

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• 3. A Syntactic Model of a Class of Mereologies
• 3. A Syntactic Model of a Class of Mereologies

3.1. Systems, Assemblies, Units

• We speak of systems as assemblies.
• From an assembly we can immediately observe a set of parts.
• Parts are either assemblies or units.
• We do not further define what assemblies and units are.

type S = A, A, U, P = A | U value

• bs Ps: (S|A) → P-set
• Parts observed from an assembly are said to be immediately embedded in, that

is, within, that assembly.

• Two or more different parts of an assembly are said to be immediately adjacent

to one another.

October 30, 2010, 15:06, Uppsala Seminar, 11 Nov. 2010 c Dines Bjørner 2010, Fredsvej 11, DK–2840 Holte, Denmark

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• 3. A Syntactic Model of a Class of Mereologies 3.1. Systems, Assemblies, Units

"outermost" Assembly A

D311 D312

C31 B3 C12 B1 Units Assemblies B4 C11 C21 C32 B2 C33 System = Environment

Figure 1: Assemblies and Units “embedded” in an Environment

• A system includes its environment.
• And we do not worry, so far, about the semiotics of all this !

c Dines Bjørner 2010, Fredsvej 11, DK–2840 Holte, Denmark Uppsala Seminar, 11 Nov. 2010 October 30, 2010, 15:06

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• 3. A Syntactic Model of a Class of Mereologies 3.1. Systems, Assemblies, Units
• Given obs Ps we can define a function, xtr Ps,

– which applies to an assembly a and – which extracts all parts embedded in a and including a.

• The functions obs Ps and xtr Ps define the meaning of embeddedness.

value xtr Ps: (S|A) → P-set xtr Ps(a) ≡ let ps = {a} ∪ obs Ps(a) in ps ∪ union{xtr Ps(a

′)|a ′:A•a ′ ∈ ps} end

• union is the distributed union operator.

October 30, 2010, 15:06, Uppsala Seminar, 11 Nov. 2010 c Dines Bjørner 2010, Fredsvej 11, DK–2840 Holte, Denmark

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• 3. A Syntactic Model of a Class of Mereologies 3.1. Systems, Assemblies, Units
• Parts have unique identifiers.
• All parts observable from a system are distinct.

type AUI value

• bs AUI: P → AUI

axiom ∀ a:A

• let ps = obs Ps(a) in

∀ p

′,p ′′:P

• {p

′,p ′′}⊆ps ∧ p ′=p ′′ ⇒ obs AUI(p ′)=obs AUI(p ′′) ∧

∀ a

′,a ′′:A

• {a

′,a ′′}⊆ps ∧ a ′=a ′′ ⇒ xtr Ps(a ′)∩ xtr Ps(a ′′)={} end

c Dines Bjørner 2010, Fredsvej 11, DK–2840 Holte, Denmark Uppsala Seminar, 11 Nov. 2010 October 30, 2010, 15:06

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• 3. A Syntactic Model of a Class of Mereologies 3.2. ‘Adjacency’ and ‘Within’ Relations

3.2. ‘Adjacency’ and ‘Within’ Relations

• Two parts, p,p′, are said to be immediately next to, i.e.,

i next to(p,p′)(a), one another in an assembly a – if there exists an assembly, a′ equal to or embedded in a – such that p and p′ are observable in that assembly a′. value i next to: P × P → A ∼ → Bool, pre i next to(p,p

′)(a): p=p ′

i next to(p,p

′)(a) ≡ ∃ a ′:A

• a

′=a ∨ a ′ ∈ xtr Ps(a)

• {p,p

′}⊆obs Ps(a ′)

October 30, 2010, 15:06, Uppsala Seminar, 11 Nov. 2010 c Dines Bjørner 2010, Fredsvej 11, DK–2840 Holte, Denmark

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• 3. A Syntactic Model of a Class of Mereologies 3.2. ‘Adjacency’ and ‘Within’ Relations
• One part, p, is said to be immediately within another part, p′in an

assembly a – if there exists an assembly, a′ equal to or embedded in a – such that p is observable in a′. value i within: P × P → A ∼ → Bool i within(p,p

′)(a) ≡

∃ a

′:A

• (a=a

′ ∨ a ′ ∈ xtr Ps(a))

• p

′=a ′ ∧ p ∈ obs Ps(a ′)

c Dines Bjørner 2010, Fredsvej 11, DK–2840 Holte, Denmark Uppsala Seminar, 11 Nov. 2010 October 30, 2010, 15:06

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• 3. A Syntactic Model of a Class of Mereologies 3.2. ‘Adjacency’ and ‘Within’ Relations
• We can generalise the immediate ‘within’ property.
• A part, p, is (transitively) within a part p′, within(p,p′)(a), of an

assembly, a, – either if p, is immediately within p′ of that assembly, a, – or if there exists a (proper) part p′′ of p′ – such that within(p′′,p)(a). value within: P × P → A ∼ → Bool within(p,p

′)(a) ≡

i within(p,p

′)(a) ∨ ∃ p ′′:P

• p

′′ ∈ obs Ps(p) ∧ within(p ′′,p ′)(a)

October 30, 2010, 15:06, Uppsala Seminar, 11 Nov. 2010 c Dines Bjørner 2010, Fredsvej 11, DK–2840 Holte, Denmark

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• 3. A Syntactic Model of a Class of Mereologies 3.2. ‘Adjacency’ and ‘Within’ Relations
• The function within can be defined, alternatively,
• using xtr Ps and i within
• instead of obs Ps and within :

value within

′: P × P → A ∼

→ Bool within

′(p,p ′)(a) ≡

i within(p,p

′)(a) ∨ ∃ p ′′:P

• p

′′ ∈ xtr Ps(p) ∧ i within(p ′′,p ′)(a)

lemma: within ≡ within

′

c Dines Bjørner 2010, Fredsvej 11, DK–2840 Holte, Denmark Uppsala Seminar, 11 Nov. 2010 October 30, 2010, 15:06

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• 3. A Syntactic Model of a Class of Mereologies 3.2. ‘Adjacency’ and ‘Within’ Relations 3.2.1. Transitive ‘Adjacency’

3.2.1. Transitive ‘Adjacency’

• We can generalise the immediate ‘next to’ property.
• Two parts, p, p′ of an assembly, a, are adjacent if they are

– either ‘next to’ one another – or if there are two parts po, p′

• ∗ such that p, p′ are embedded in respectively po and p′
• ∗ and such that po, p′
• are immediately next to one another.

October 30, 2010, 15:06, Uppsala Seminar, 11 Nov. 2010 c Dines Bjørner 2010, Fredsvej 11, DK–2840 Holte, Denmark

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• 3. A Syntactic Model of a Class of Mereologies 3.2. ‘Adjacency’ and ‘Within’ Relations 3.2.1. Transitive ‘Adjacency’

value adjacent: P × P → A ∼ → Bool adjacent(p,p

′)(a) ≡

i next to(p,p

′)(a) ∨

∃ p

′′,p ′′′:P

• {p

′′,p ′′′}⊆xtr Ps(a) ∧ i next to(p ′′,p ′′′)(a) ∧

((p=p

′′)∨within(p,p ′′)(a)) ∧ ((p ′=p ′′′)∨within(p ′,p ′′′)(a))

c Dines Bjørner 2010, Fredsvej 11, DK–2840 Holte, Denmark Uppsala Seminar, 11 Nov. 2010 October 30, 2010, 15:06

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• 3. A Syntactic Model of a Class of Mereologies 3.3. Mereology, Part I

3.3. Mereology, Part I

• So far we have built a ground mereology model, MGround.
• Let ⊑ denote parthood, x is part of y, x ⊑ y.

∀x(x ⊑ x)1 (1) ∀x, y(x ⊑ y) ∧ (y ⊑ x) ⇒ (x = y) (2) ∀x, y, z(x ⊑ y) ∧ (y ⊑ z) ⇒ (x ⊑ z) (3)

• Let ❁ denote proper parthood, x is part of y, x ❁ y.
• Formula 4 defines x ❁ y. Equivalence 5 can be proven to hold.

∀x ❁ y =def x(x ⊑ y) ∧ ¬(x = y) (4) ∀∀x, y(x ⊑ y) ⇔ (x ❁ y) ∨ (x = y) (5)

1Our notation now is not RSL but some conventional first-order predicate logic notation.

October 30, 2010, 15:06, Uppsala Seminar, 11 Nov. 2010 c Dines Bjørner 2010, Fredsvej 11, DK–2840 Holte, Denmark

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[ 3. A Syntactic Model of a Class of Mereologies, 3.3. Mereology, Part I ]

• The proper part (x ❁ y) relation is a strict partial ordering:

∀x¬(x ❁ x) (6) ∀x, y(x ❁ y) ⇒ ¬(y ❁ x) (7) ∀x, y, z(x ❁ y) ∧ (y ❁ z) ⇒ (x ❁ z) (8)

• Overlap, •, is also a relation of parts:

– Two individuals overlap if they have parts in common: x • y =def ∃z(z ❁ x) ∧ (z ❁ y) (9) ∀x(x • x) (10) ∀x, y(x • y) ⇒ (y • x) (11)

c Dines Bjørner 2010, Fredsvej 11, DK–2840 Holte, Denmark Uppsala Seminar, 11 Nov. 2010 October 30, 2010, 15:06

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• 3. A Syntactic Model of a Class of Mereologies 3.3. Mereology, Part I
• Proper overlap, ◦, can be defined:

x ◦ y =def (x • x) ∧ ¬(x ⊑ y) ∧ ¬(y ⊑ x) (12)

• Whereas Formulas (1-11) holds of the model of mereology we have

shown so far, Formula (12) does not.

• In the next section we shall repair that situation.
• The proper part relation, ❁, reflects the within relation.
• The disjoint relation,
• , reflects the adjacency relation.

x

• y =def ¬(x • y)

(13)

October 30, 2010, 15:06, Uppsala Seminar, 11 Nov. 2010 c Dines Bjørner 2010, Fredsvej 11, DK–2840 Holte, Denmark

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• 3. A Syntactic Model of a Class of Mereologies 3.3. Mereology, Part I
• Disjointness is symmetric:

∀x, y(x

• y) ⇒ (y
• x)

(14)

• The weak supplementation relation, Formula 15, expresses

– that if y is a proper part of x – then there exists a part z – such that z is a proper part of x – and z and y are disjoint

• That is, whenever an individual has one proper part then it has

more than one. ∀x, y(y ❁ x) ⇒ ∃z(z ❁ x) ∧ (z

• y)

(15)

c Dines Bjørner 2010, Fredsvej 11, DK–2840 Holte, Denmark Uppsala Seminar, 11 Nov. 2010 October 30, 2010, 15:06

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• 3. A Syntactic Model of a Class of Mereologies 3.3. Mereology, Part I
• Formulas 1–3 and 15 together determine the minimal mereology,

MMinimal.

• Formula 15 does not hold of the model of mereology we have shown

so far.

• We shall comment on this once we have introduced the notion of of

parts having attributes.

October 30, 2010, 15:06, Uppsala Seminar, 11 Nov. 2010 c Dines Bjørner 2010, Fredsvej 11, DK–2840 Holte, Denmark

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• 3. A Syntactic Model of a Class of Mereologies 3.4. A Syntactic Model of a Class of Mereologies

3.4. A Syntactic Model of a Class of Mereologies 3.4.1. Connectors

• So far we have only covered notions of

– parts being next to other parts or – within one another.

• We shall now add to this a rather general notion of parts being
• therwise related.
• That notion is one of connectors.

c Dines Bjørner 2010, Fredsvej 11, DK–2840 Holte, Denmark Uppsala Seminar, 11 Nov. 2010 October 30, 2010, 15:06

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• 3. A Syntactic Model of a Class of Mereologies 3.4. A Syntactic Model of a Class of Mereologies 3.4.1. Connectors
• Connectors provide for connections between parts.
• A connector is an ability be be connected.
• A connection is the actual fulfillment of that ability.
• Connections are relations between pairs of parts.
• Connections “cut across” the “classical”

– parts being part of the (or a) whole and – parts being related by embeddedness or adjacency.

October 30, 2010, 15:06, Uppsala Seminar, 11 Nov. 2010 c Dines Bjørner 2010, Fredsvej 11, DK–2840 Holte, Denmark

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• 3. A Syntactic Model of a Class of Mereologies 3.4. A Syntactic Model of a Class of Mereologies 3.4.1. Connectors

A

D311 D312

C31 B3 C12 B1 Units Assemblies B4 C11 C21 C32 "outermost" Assembly

K2

B2 C33

K1

System = Environment

Figure 2: Assembly and Unit Connectors: Internal and External

• For now, we do not “ask” for the meaning of connectors !

c Dines Bjørner 2010, Fredsvej 11, DK–2840 Holte, Denmark Uppsala Seminar, 11 Nov. 2010 October 30, 2010, 15:06

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• 3. A Syntactic Model of a Class of Mereologies 3.4. A Syntactic Model of a Class of Mereologies 3.4.1. Connectors
• Figure 2 on the facing page “adds” connectors to Fig. 1 on page 14.
• The idea is that connectors

– allow an assembly to be connected to any embedded part, and – allow two adjacent parts to be connected.

• In Fig. 2 on the facing page

– the environment is connected, by K2, to part C11; – the “external world” is connected, by K1, to B1; – etcetera.

October 30, 2010, 15:06, Uppsala Seminar, 11 Nov. 2010 c Dines Bjørner 2010, Fredsvej 11, DK–2840 Holte, Denmark

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• 3. A Syntactic Model of a Class of Mereologies 3.4. A Syntactic Model of a Class of Mereologies 3.4.1. Connectors
• From a system we can observe all its connectors.
• From a connector we can observe

– its unique connector identifier and – the set of part identifiers of the parts that the connector connects.

• All part identifiers of system connectors identify parts of the system.
• All observable connector identifiers of parts identify connectors of

the system.

c Dines Bjørner 2010, Fredsvej 11, DK–2840 Holte, Denmark Uppsala Seminar, 11 Nov. 2010 October 30, 2010, 15:06

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• 3. A Syntactic Model of a Class of Mereologies 3.4. A Syntactic Model of a Class of Mereologies 3.4.1. Connectors

type K value

• bs Ks: S → K-set
• bs KI: K → KI
• bs Is: K → AUI-set
• bs KIs: P → KI-set

axiom ∀ k:K

• card obs Is(k)=2,

∀ s:S,k:K

• k ∈ obs Ks(s) ⇒

∃ p:P

• p ∈ xtr Ps(s) ⇒ obs AUI(p) ∈ obs Is(k),

∀ s:S,p:P

• ∀ ki:KI
• ki ∈ obs KIs(p) ⇒

∃! k:K

• k ∈ obs Ks(s) ∧ ki=obs KI(k)

October 30, 2010, 15:06, Uppsala Seminar, 11 Nov. 2010 c Dines Bjørner 2010, Fredsvej 11, DK–2840 Holte, Denmark

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• 3. A Syntactic Model of a Class of Mereologies 3.4. A Syntactic Model of a Class of Mereologies 3.4.1. Connectors
• This model allows for a rather “free-wheeling” notion of connectors

– one that allows internal connectors to “cut across” embedded and adjacent parts; – and one that allows external connectors to “penetrate” from an

• utside to any embedded part.

c Dines Bjørner 2010, Fredsvej 11, DK–2840 Holte, Denmark Uppsala Seminar, 11 Nov. 2010 October 30, 2010, 15:06

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• 3. A Syntactic Model of a Class of Mereologies 3.4. A Syntactic Model of a Class of Mereologies 3.4.1. Connectors
• We need define an auxiliary function.

– xtr∀KIs(p) applies to a system – and yields all its connector identifiers. value xtr∀KIs: S → KI-set xtr∀Ks(s) ≡ {obs KI(k)|k:K•k ∈ obs Ks(s)}

October 30, 2010, 15:06, Uppsala Seminar, 11 Nov. 2010 c Dines Bjørner 2010, Fredsvej 11, DK–2840 Holte, Denmark

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• 3. A Syntactic Model of a Class of Mereologies 3.5. Mereology, Part II

3.5. Mereology, Part II

We shall interpret connections as follows:

• A connection between parts pi and pj

– that enjoy a pi adjacent to pj relationship, means pi ◦ pj, – that is, although parts pi and pj are adjacent – they do share “something”, i.e., have something in common. – What that “something” is we shall comment on later, when we have “mapped” systems onto parallel compositions of CSP processes.

• A connection between parts pi and pj

– that enjoy a pi within pj relationship, – does not add other meaning than – commented upon later, again when we have “mapped” systems

• nto parallel compositions of CSP processes.

c Dines Bjørner 2010, Fredsvej 11, DK–2840 Holte, Denmark Uppsala Seminar, 11 Nov. 2010 October 30, 2010, 15:06

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• 3. A Syntactic Model of a Class of Mereologies 3.5. Mereology, Part II
• With the above interpretation we may arrive at the following,

perhaps somewhat “awkward-looking” case: – a connection connects two adjacent parts pi and pj ∗ where part pi is within part pio ∗ and part pj is within part pjo ∗ where parts pio and pjo are adjacent ∗ but not otherwise connected. – How are we to explain that ! ∗ Since we have not otherwise interpreted the meaning of parts, ∗ we can just postulate that “so it is” ! ∗ We shall, later, again when we have “mapped” systems onto parallel compositions of CSP processes, give a more satisfactory explanation.

October 30, 2010, 15:06, Uppsala Seminar, 11 Nov. 2010 c Dines Bjørner 2010, Fredsvej 11, DK–2840 Holte, Denmark

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• 3. A Syntactic Model of a Class of Mereologies 3.5. Mereology, Part II
• We earlier introduced the following operators:

– ⊑, ❁, •, ◦, and

• In some of the mereology literature [BowmanLClarke81,

BowmanLClarke85, CasatiVarzi1999] these operators are symbolised with caligraphic letters: – ⊑: P: part, – ❁: PP: proper part, – • : O: overlap and –

• : U: underlap.

c Dines Bjørner 2010, Fredsvej 11, DK–2840 Holte, Denmark Uppsala Seminar, 11 Nov. 2010 October 30, 2010, 15:06

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• 4. Discussion & Interpretation
• 4. Discussion & Interpretation
• Before a semantic treatment of the concept of mereology

– let us review what we have done and – let us interpret our abstraction ∗ (i.e., relate it to actual societal infrastructure components).

October 30, 2010, 15:06, Uppsala Seminar, 11 Nov. 2010 c Dines Bjørner 2010, Fredsvej 11, DK–2840 Holte, Denmark

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[ 4. Discussion & Interpretation ]

4.1. What We have Done So Far ?

• We have

– presented a model that is claimed to abstract essential mereological properties of ∗ machine assemblies, ∗ railway nets, ∗ the oil industry, ∗ oil pipelines, ∗ buildings with installations, ∗ hospitals, ∗ etcetera.

c Dines Bjørner 2010, Fredsvej 11, DK–2840 Holte, Denmark Uppsala Seminar, 11 Nov. 2010 October 30, 2010, 15:06

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• 4. Discussion & Interpretation 4.2. Six Interpretations

4.2. Six Interpretations

• Let us substantiate the claims made in the previous paragraph.

– We will do so, albeit informally, in the next many paragraphs. – Our substantiation is a form of diagrammatic reasoning. – Subsets of diagrams will be claimed to represent parts, while – Other subsets will be claimed to represent connectors.

• The reasoning is incomplete.

October 30, 2010, 15:06, Uppsala Seminar, 11 Nov. 2010 c Dines Bjørner 2010, Fredsvej 11, DK–2840 Holte, Denmark

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• 4. Discussion & Interpretation 4.2. Six Interpretations 4.2.1. Air Traffic

4.2.1. Air Traffic

Ground Control Tower Aircraft Control Tower Continental Control Control Control Control Control Continental Tower Tower Ground Control

1..k..t 1..m..r 1..n..c 1..n..c 1..j..a 1..i..g 1..m..r 1..k..t 1..i..g

This right 1/2 is a "mirror image" of left 1/2 of figure

ac/ca[k,n]:AC|CA cc[n,n’]:CC rc/cr[m,n]:RC|CR ac/ca[k,n]:AC|CA rc/cr[m,n]:RC|CR ga/ag[i,j]:GA|AG ga/ag[i,j]:GA|AG at/ta[k,j]:AT|TA at/ta[k,j]:AT|TA gc/cg[i,n]:GC|CG ar/ra[m,j]:AR|RA ar/ra[m,j]:AR|RA gc/cg[i,n]:GC|CG

Terminal Terminal Area Area Centre Centre Centre Centre

Figure 3: An air traffic system. Black boxes and lines are units; red boxes are connections

c Dines Bjørner 2010, Fredsvej 11, DK–2840 Holte, Denmark Uppsala Seminar, 11 Nov. 2010 October 30, 2010, 15:06

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• 4. Discussion & Interpretation 4.2. Six Interpretations 4.2.2. Buildings

4.2.2. Buildings

A H I J L M K C F G E B D

Door Connector Door Connection Installation Connector

(1 Unit)

Installation Room

(1 Unit)

Sub−room of Room Sharing walls

(1 Unit)

Adjacent Rooms Sharing (one) wall

(2 Units)

κ γ ε ι ω

Figure 4: A building plan with installation

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• 4. Discussion & Interpretation 4.2. Six Interpretations 4.2.3. Financial Service Industry

4.2.3. Financial Service Industry

Clients C[c] C[2] C[1] T[1] T[2] T[1]

cb/bc[1..c,1..b]:CB|BC ct/tc[1..c,1..t]:CT|TC cp/pc[1..c,1..p]:CP|PC bt/tb[1..b,1..t]:BT|TB pt/tp[1..p,1..t]:PT|TP pb/bp[1..p,1..b]:PB|BP

The Finance Industry "Watchdog"

wb/bw[1..b]:WB|BW wt/tw[1..t]:WT|TW wp/pw[1..p]:WP|PW ws:WS sw:SW

SE Exchange Stock

I[1] I[1] I[2] I[i]

... ...

is/si[1..i]:IS|SI

B[1] B[2] B[b]

...

Banks P[1] P[2] P[p]

...

Portfolio Managers

...

Brokers Traders Figure 5: A financial service industry c Dines Bjørner 2010, Fredsvej 11, DK–2840 Holte, Denmark Uppsala Seminar, 11 Nov. 2010 October 30, 2010, 15:06

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• 4. Discussion & Interpretation 4.2. Six Interpretations 4.2.4. Machine Assemblies

4.2.4. Machine Assemblies

Connection Connector, part of Connection Connector, part of Connection Connection Part Assembly, embedded Part Adjacent Parts Bellows Coil/ Air Load Reservoir Valve1 with one Unit with two Assembly System Assembly Assembly Valve2 Unit Unit Unit Unit Unit Unit Unit Units Magnet Pump Power Supply Air Supply Lever Unit Unit 2 Parts, one Assembly with is an Assembly

Figure 6: An air pump, i.e., a physical mechanical system

October 30, 2010, 15:06, Uppsala Seminar, 11 Nov. 2010 c Dines Bjørner 2010, Fredsvej 11, DK–2840 Holte, Denmark

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• 4. Discussion & Interpretation 4.2. Six Interpretations 4.2.5. Oil Industry

4.2.5. Oil Industry

4.2.5.1. “The” Overall Assembly

Oil Field Pipeline System Refinery Port Port Ocean Port Port Port Distrib. Distrib. Distrib. Refinery Distrib. Assembly Connection (bound) Connection (free)

Figure 7: A Schematic of an Oil Industry

c Dines Bjørner 2010, Fredsvej 11, DK–2840 Holte, Denmark Uppsala Seminar, 11 Nov. 2010 October 30, 2010, 15:06

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• 4. Discussion & Interpretation 4.2. Six Interpretations 4.2.5. Oil Industry 4.2.5.2. A Concretised Assembly Unit

4.2.5.2. A Concretised Assembly Unit

fpb vz vx fpa fpc vw fpd vu vy p1 p2 p3 p4 p5 p7 p6 p10 p11 p12 p8 p9 p13 p14 p15 inj inl

• nr
• ns

Connector Node unit Connection (between pipe units and node units) Pipe unit

ini ink

may connect to refinery

• np
• nq

may be left "dangling" may be left dangling may connect to oil field

Figure 8: A Pipeline System

October 30, 2010, 15:06, Uppsala Seminar, 11 Nov. 2010 c Dines Bjørner 2010, Fredsvej 11, DK–2840 Holte, Denmark

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• 4. Discussion & Interpretation 4.2. Six Interpretations 4.2.6. Railway Nets

4.2.6. Railway Nets

Turnout / Point Track / Line / Segment / Linear Unit / Switch Unit / Rigid Crossing Switchable Crossover Unit / Double Slip Connectors − in−between are Units Simple Crossover Unit

Figure 9: Four example rail units

c Dines Bjørner 2010, Fredsvej 11, DK–2840 Holte, Denmark Uppsala Seminar, 11 Nov. 2010 October 30, 2010, 15:06

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• 4. Discussion & Interpretation 4.2. Six Interpretations 4.2.6. Railway Nets

Connector Connection Linear Unit Switch Track Siding

Station

Switchable Crossover Line

Station

Crossover

Figure 10: A “model” railway net. An Assembly of four Assemblies: Two stations and two lines; Lines here consist of linear rail units; stations of all the kinds of units shown in Fig. 9 on the facing page. There are 66 connections and four “dangling” connectors

October 30, 2010, 15:06, Uppsala Seminar, 11 Nov. 2010 c Dines Bjørner 2010, Fredsvej 11, DK–2840 Holte, Denmark

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• 4. Discussion & Interpretation 4.3. Discussion

4.3. Discussion

• It requires a somewhat more laborious effort,

– than just “flashing” and commenting on these diagrams, – to show that the modelling of essential aspects of their structures – can indeed be done by simple instantiation – of the model given in the previous part of the talk.

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[ 4. Discussion & Interpretation, 4.3. Discussion ]

• We can refer to a number of documents which give rather detailed

domain models of – air traffic, – container line industry, – financial service industry, – health-care, – IT security, – “the market”, – “the” oil industry2, – transportation nets3, – railways, etcetera, etcetera.

• Seen in the perspective of the present paper

– we claim that much of the modelling work done in those references – can now be considerably shortened and – trust in these models correspondingly increased.

2http://www2.imm.dtu.dk/˜db/pipeline.pdf 3http://www2.imm.dtu.dk/˜db/transport.pdf

October 30, 2010, 15:06, Uppsala Seminar, 11 Nov. 2010 c Dines Bjørner 2010, Fredsvej 11, DK–2840 Holte, Denmark

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• 4. Discussion & Interpretation 4.4. Mereology, Part III

4.4. Mereology, Part III

• Formula 15 on page 26 expresses that

– whenever an individual has one proper part – then it has more than one.

• We mentioned there, Slide 27, that we would comment on the fact

that our model appears to allow that assemblies may have just one proper part.

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• 4. Discussion & Interpretation 4.4. Mereology, Part III
• We now do so.

– We shall still allow assemblies to have just one proper part — – in the sense of a sub-assembly or a unit — – but we shall interpret the fact that an assembly always have at least one attribute. – Therefore we shall “generously” interpret the set of attributes of an assembly to constitute a part.

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• 4. Discussion & Interpretation 4.4. Mereology, Part III
• In Sect. 5

– we shall see how attributes of both units and assemblies of the interpreted mereology – contribute to the state components of the unit and assembly processes.

c Dines Bjørner 2010, Fredsvej 11, DK–2840 Holte, Denmark Uppsala Seminar, 11 Nov. 2010 October 30, 2010, 15:06

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[ 4. Simple Entities ]

4.5. Discussion

• In Sect. 3.2 we interpreted the model of mereology in six examples.
• The units of Sect. 2

– which in that section were left uninterpreted – now got individuality — ∗ in the form of · aircraft, · building rooms, · rail units and · oil pipes. – Similarly for the assemblies of Sect. 2. They became ∗ pipeline systems, ∗ oil refineries, ∗ train stations, ∗ banks, etc.

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[ 4. Simple Entities, 4.5. Discussion ]

• In conventional modelling

– the mereology of an infrastructure component, ∗ of the kinds exemplified in Sect. 3.2, – was modelled by modelling ∗ that infrastructure component’s special mereology ∗ together, “in line”, with the modelling ∗ of unit and assembly attributes.

• With the model of Sect. 2 now available

– we do not have to model the mereological aspects, – but can, instead, instantiate the model of Sect. 2 appropriately. – We leave that to be reported upon elsewhere.

c Dines Bjørner 2010, Fredsvej 11, DK–2840 Holte, Denmark Uppsala Seminar, 11 Nov. 2010 October 30, 2010, 15:06

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• 5. A Semantic Model of a Class of Mereologies
• 5. A Semantic Model of a Class of Mereologies

5.1. The Mereology Entities ≡ Processes

• The model of mereology (Slides 13–38) given earlier focused on the

following simple entities (i) the assemblies, (ii) the units and (iii) the connectors.

• To assemblies and units we associate CSP processes, and
• to connectors we associate a CSP channels,
• one-by-one.
• The connectors form the mereological attributes of the model.

October 30, 2010, 15:06, Uppsala Seminar, 11 Nov. 2010 c Dines Bjørner 2010, Fredsvej 11, DK–2840 Holte, Denmark

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• 5. A Semantic Model of a Class of Mereologies 5.1. The Mereology Entities ≡ Processes 5.1.1. Channels

5.1.1. Channels

• The CSP channels,

– are each “anchored” in two parts: – if a part is a unit then in “its corresponding” unit process, and – if a part is an assembly then in “its corresponding” assembly process.

• From a system assembly we can extract all connector identifiers.
• They become indexes into an array of channels.

– Each of the connector channel identifiers is mentioned – in exactly two unit or assembly processes.

c Dines Bjørner 2010, Fredsvej 11, DK–2840 Holte, Denmark Uppsala Seminar, 11 Nov. 2010 October 30, 2010, 15:06

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• 5. A Semantic Model of a Class of Mereologies 5.1. The Mereology Entities ≡ Processes 5.1.1. Channels

value s:S kis:KI-set = xtr∀KIs(s) type ChMap = AUI →

m KI-set

value cm:ChMap = [ obs AUI(p)→obs KIs(p)|p:P•p ∈ xtr Ps(s) ] channel ch[ i|i:KI•i ∈ kis ] MSG

October 30, 2010, 15:06, Uppsala Seminar, 11 Nov. 2010 c Dines Bjørner 2010, Fredsvej 11, DK–2840 Holte, Denmark

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• 5. A Semantic Model of a Class of Mereologies 5.2. Process Definitions

5.2. Process Definitions value system: S → Process system(s) ≡ assembly(s) assembly: a:A→in,out {ch[ cm(i) ]|i:KI•i ∈ cm(obs AUI(a))} process assembly(a) ≡ MA(a)(obs AΣ(a)) {assembly(a

′)|a ′:A•a ′ ∈ obs Ps(a)}

{unit(u)|u:U•u ∈ obs Ps(a)}

• bs AΣ: A → AΣ

MA: a:A→AΣ→in,out {ch[ cm(i) ]|i:KI•i ∈ cm(obs AUI(a))} process MA(a)(aσ) ≡ MA(a)(AF(a)(aσ)) AF: a:A → AΣ → in,out {ch[ em(i) ]|i:KI•i ∈ cm(obs AUI(a))}×AΣ

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• 5. A Semantic Model of a Class of Mereologies 5.2. Process Definitions

unit: u:U → in,out {ch[ cm(i) ]|i:KI•i ∈ cm(obs UI(u))} process unit(u) ≡ MU(u)(obs UΣ(u))

• bs UΣ: U → UΣ

MU: u:U → UΣ → in,out {ch[ cm(i) ]|i:KI•i ∈ cm(obs UI(u))} process MU(u)(uσ) ≡ MU(u)(UF(u)(uσ)) UF: U → UΣ → in,out {ch[ em(i) ]|i:KI

• i ∈ cm(obs AUI(u))} UΣ

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• 5. A Semantic Model of a Class of Mereologies 5.3. Mereology, Part III

5.3. Mereology, Part III

• A little more meaning has been added to the notions of parts and

connections.

• The within and adjacent to relations between parts (assemblies and

units) reflect a phenomenological world of geometry, and

• the connected relation between parts (assemblies and units)

– reflect both physical and conceptual world understandings: ∗ physical world in that, for example, radio waves cross geometric “boundaries”, and ∗ conceptual world in that ontological classifications typically reflect lattice orderings where overlaps likewise cross geometric “boundaries”.

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• 5. A Semantic Model of a Class of Mereologies 5.4. Discussion

5.4. Discussion

• That completes our ‘contribution’:

– A mereology of systems has been given – a syntactic explanation, Sect. 2, – a semantic explanation, Sect. 5 and – their relationship to classical mereologies.

October 30, 2010, 15:06, Uppsala Seminar, 11 Nov. 2010 c Dines Bjørner 2010, Fredsvej 11, DK–2840 Holte, Denmark

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• 5. A Semantic Model of a Class of Mereologies 5.5. Acknowledgements

5.5. Acknowledgements

• I thank Lars-Henrik Eriksson and his colleagues for inviting me to

give a two week “PhD School” series of lectures at Uppsala.

• And I also thank for the opportunity to give this seminar.

c Dines Bjørner 2010, Fredsvej 11, DK–2840 Holte, Denmark Uppsala Seminar, 11 Nov. 2010 October 30, 2010, 15:06